Problem: Integrate. $\int\left(5e^x-\dfrac2x \right)dx=\,?$ Choose 1 answer: Choose 1 answer: (Choice A) A $e^{5x}-2\ln|x|+C$ (Choice B) B $5e^x-2\ln|x|+C$ (Choice C) C $e^{5x}-\ln|2|+C$ (Choice D) D $5e^x-\ln|2|+C$
Solution: We can integrate using the following formulas for the indefinite integrals of $e^x$ and $\dfrac1x$ : $\begin{aligned} &\int e^x\,dx=e^x+C \\\\ &\int \dfrac1x\,dx=\ln|x|+C \end{aligned}$ $\begin{aligned} &\phantom{=}\int\left(5e^x-\dfrac2x \right)dx \\\\ &=5\int e^x\,dx-2\int\dfrac1x \,dx \\\\ &=5e^x-2\ln|x|+C \end{aligned}$